A 4^th-order conservative advection scheme on triangular meshes using Hermite interpolation

This paper presents a high order conservative scheme on triangular meshes for 2D hyperbolic equations. In this scheme, a bivariate cubic Hermite interpolation polynomial is constructed on each triangular element, using the function value and its derivatives at the vertexes as well as the cell-integrated average at the present moment as interpolation conditions. Then the physical quantities on the vertexes at the next time step are predicted via a Semi-Lagrangian solution procedure. Meanwhile, in order to preserve the conservation, the cell-integrated average is updated by using finite volume method. The present scheme extends the original CIP method to unstructured meshes and makes it more flexible to solve more complex boundary problems. The scheme is explicit, compact, simple and easy to be implemented. Numerical experiments for smooth problems are carried out to demonstrate 4th-order spatial accuracy of this scheme, just the same as that of the original CIP method. Moreover, this scheme can accurately capture shocks when applied to non-smooth problems, which indicates its conservativeness.